Efficient deflection angle based filtering for waveform inversion

The importance of accessing the scattering angle information has been recognized in ray-based applications a long time ago, but only recently became available in the field of wave equation based imaging and inversion. First it was implemented for wave equation migration velocity analysis, then reverse-time migration and finally full-waveform inversion. Conventional access to the scattering angle information in seismic imaging via wavefield continuation requires an extension either in space or in time, which is costly in terms of computational resources. For a single frequency this filtering can however be interpreted as a non-stationary convolutional filtering, which is expensive in general, but more so in 3D models. To obtain a more efficient scattering angle filter, we develop techniques that utilize the mapping nature (no domain extension) of the scattering angle based filter for constant-velocity background models. We split the background velocity model into regions with different velocity ranges, generating an "extension in velocity", so that in each region the velocity is assumed not to vary much. A numerical example demonstrates that a few samples in the newly introduced dimension is enough to apply the scattering angle filter. The filter can be utilized either for full-waveform inversion preconditioning or to clean up reverse-time migration artifacts. A novel interpolation is obtained by splitting the background velocity model with a smooth decomposition of unity.